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In the search for experimental realizations of bond-anisotropic Kitaevinteractions and resulting spin-liquid phases, the layered magnet$\alpha$-RuCl$_3$ is a prime candidate. Its modelling typically involvesHeisenberg, Kitaev, and symmetric off-diagonal $\Gamma$ interactions on thetwo-dimensional honeycomb lattice. However, recent neutron-scatteringexperiments point towards a sizeable magnetic interlayer coupling. Here westudy three-dimensional exchange models for $\alpha$-RuCl$_3$, for bothpossible $R\bar{3}$ and $C2/m$ crystal structures. We discuss the symmetryconstraints on the interlayer couplings, construct minimal models, and use themto compute the magnetic mode dispersion and the dynamical spin structurefactor, in both the zero-field zigzag phase and the paramagnetic high-fieldphase. Our predictions for the interlayer mode dispersion shall guide futureexperiments; they also call for a reevaluation of the quantitative modelparameters relevant for $\alpha$-RuCl$_3$.

Magnon dispersion and dynamic spin response in three-dimensional spin models for $\alpha$-RuCl$_3$